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RD Sharma solutions for Class 12 Mathematics chapter 11 - Differentiation

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Chapter 11 : Differentiation

Page 17

Q 1 | Page 17

Differentiate the following functions from first principles e−x.

Q 2 | Page 17

Differentiate the following functions from first principles e3x.

Q 3 | Page 17

Differentiate the following functions from first principles eax+b.

Q 4 | Page 17

Differentiate the following functions from first principles ecos x.

Q 5 | Page 17

Differentiate the following functions from first principles  \[e^\sqrt{2x}\].

Q 6 | Page 17

Differentiate the following functions from first principles log cos x ?

Q 7 | Page 17

​Differentiate the following function from first principles \[e^\sqrt{\cot x}\] .

Q 8 | Page 17

Differentiate the following functions from first principles x2ex ?

Q 9 | Page 17

Differentiate the following functions from first principles log cosec x ?

Q 10 | Page 17

Differentiate the following functions from first principles sin−1 (2x + 3) ?

Pages 37 - 38

Q 1 | Page 37

Differentiate sin (3x + 5) ?

Q 2 | Page 37

Differentiate tan2 x ?

Q 3 | Page 37

Differentiate tan (x° + 45°) ?

Q 4 | Page 37

Differentiate sin (log x) ?

Q 5 | Page 37

Differentiate \[e^{\sin} \sqrt{x}\] ?

Q 6 | Page 37

Differentiate etan x ?

Q 7 | Page 37

Differentiate sin2 (2x + 1) ?

Q 8 | Page 37

Differentiate log7 (2x − 3) ?

Q 9 | Page 37

Differentiate tan 5x° ?

Q 10 | Page 37

Differentiate \[{2^x}^3\] ?

Q 11 | Page 37

Differentiate \[3^{e^x}\] ?

Q 12 | Page 37

Differentiate logx 3 ?

Q 13 | Page 37

Differentiate \[3^{x^2 + 2x}\] ?

Q 14 | Page 37

Differentiate \[\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\] ?

Q 15 | Page 37

Differentiate \[3^{x \log x}\] ?

Q 16 | Page 37

Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?

Q 17 | Page 37

Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?

Q 18 | Page 37

Differentiate (log sin x)?

Q 19 | Page 37

Differentiate \[\sqrt{\frac{1 + x}{1 - x}}\] ?

Q 20 | Page 37

Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?

Q 22 | Page 37

Differentiate \[e^{3 x} \cos 2x\] ?

Q 23 | Page 37

Differentiate \[e^{\tan 3 x} \] ?

Q 24 | Page 37

Differentiate \[e^\sqrt{\cot x}\] ?

Q 25 | Page 37

Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?

Q 26 | Page 37

Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?

Q 27 | Page 37

Differentiate \[\tan \left( e^{\sin x }\right)\] ?

Q 28 | Page 37

Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?

Q 29 | Page 37

Differentiate \[\frac{e^x \log x}{x^2}\] ? 

Q 30 | Page 37

Differentiate \[\log \left( cosec x - \cot x \right)\] ?

Q 31 | Page 37

Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?

Q 32 | Page 37

Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?

Q 33 | Page 37

Differentiate \[\tan^{- 1} \left( e^x \right)\] ?

Q 34 | Page 37

Differentiate \[e^{\sin^{- 1} 2x}\] ?

Q 35 | Page 37

Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?

Q 36 | Page 37

Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?

Q 37 | Page 37

Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?

Q 38 | Page 37

Differentiate \[\log \left( \tan^{- 1} x \right)\]? 

Q 39 | Page 37

Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\] ?

Q 40 | Page 37

Differentiate \[x \sin 2x + 5^x + k^k + \left( \tan^2 x \right)^3\] ?

Q 41 | Page 37

Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?

Q 42 | Page 37

Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?

Q 43 | Page 37

Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?

Q 44 | Page 37

Differentiate  \[e^x \log \sin 2x\] ?

Q 46 | Page 37

Differentiate \[\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}\] ?

Q 47 | Page 37

Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?

Q 48 | Page 37

Differentiate \[\sin^{- 1} \left( \frac{x}{\sqrt{x^2 + a^2}} \right)\] ?

Q 49 | Page 37

Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?

Q 50 | Page 37

Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?

Q 51 | Page 37

Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?

Q 52 | Page 38

Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?

Q 53 | Page 38

\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?

Q 54 | Page 38

Differentiate \[e^{ax} \sec x \tan 2x\] ?

Q 55 | Page 38

Differentiate \[\log \left( \cos x^2 \right)\] ?

Q 56 | Page 38

Differentiate \[\cos \left( \log x \right)^2\] ?

Q 57 | Page 38

Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?

Q 58 | Page 38

If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?

Q 59 | Page 38

 If \[y = \sqrt{x + 1} + \sqrt{x - 1}\] , prove that \[\sqrt{x^2 - 1}\frac{dy}{dx} = \frac{1}{2}y\] ?

Q 60 | Page 38

If \[y = \frac{x}{x + 2}\]  , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ? 

Q 61 | Page 38

If \[y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]prove that \[\frac{dy}{dx} = \frac{x - 1}{2x \left( x + 1 \right)}\] ?

 

Q 62 | Page 38

If  \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\]  prove that \[\frac{dy}{dx} = \sec 2x\] ?

Q 63 | Page 38

If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\], prove that  \[2 x\frac{dy}{dx} = \sqrt{x} - \frac{1}{\sqrt{x}}\] ?

Q 64 | Page 38

If \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] ,  prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?

Q 65 | Page 38

If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?

Q 66 | Page 38

If  \[y = \left( x - 1 \right) \log \left( x - 1 \right) - \left( x + 1 \right) \log \left( x + 1 \right)\] , prove that \[\frac{dy}{dc} = \log \left( \frac{x - 1}{1 + x} \right)\] ?

Q 67 | Page 38

If \[y = e^x \cos x\] ,prove that \[\frac{dy}{dx} = \sqrt{2} e^x \cdot \cos \left( x + \frac{\pi}{4} \right)\] ?

Q 68 | Page 38

If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 cosec 2x \] ?

Q 69 | Page 38

If \[y = x \sin^{- 1} x + \sqrt{1 - x^2}\] ,prove that \[\frac{dy}{dx} = \sin^{- 1} x\] ?

Q 70 | Page 38

If \[y = \sqrt{x^2 + a^2}\] prove that  \[y\frac{dy}{dx} - x = 0\] ?

Q 71 | Page 38

If \[y = e^x + e^{- x}\] prove that  \[\frac{dy}{dx} = \sqrt{y^2 - 4}\] ?

Q 72 | Page 38

If \[y = \sqrt{a^2 - x^2}\] prove that  \[y\frac{dy}{dx} + x = 0\] ?

Q 73 | Page 38

If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?

Q 74 | Page 38

Prove that \[\frac{d}{dx} \left\{ \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{- 1} \frac{x}{a} \right\} = \sqrt{a^2 - x^2}\] ?

Pages 62 - 64

Q 1 | Page 62

Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?

Q 2 | Page 62

Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?

Q 3 | Page 63

Differentiate  \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\]  ?

Q 4 | Page 63

Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?

Q 5 | Page 63

Differentiate \[\tan^{- 1} \left\{ \frac{x}{\sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?

Q 6 | Page 63

Differentiate \[\sin^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?

Q 7 | Page 63

Differentiate \[\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1\]  ?

Q 8 | Page 63

Differentiate \[\sin^{- 1} \left( 1 - 2 x^2 \right), 0 < x < 1\] ?

Q 9 | Page 63

Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?

Q 10 | Page 63

Differentiate \[\sin^{- 1} \left\{ \frac{\sin x + \cos x}{\sqrt{2}} \right\}, - \frac{3 \pi}{4} < x < \frac{\pi}{4}\] ?

Q 11 | Page 63

Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?

Q 12 | Page 63

Differentiate \[\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1\] ?

Q 13 | Page 63

Differentiate \[\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?

Q 14 | Page 63

Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?

Q 15 | Page 63

Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?

Q 16 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{4x}{1 - 4 x^2} \right), - \frac{1}{2} < x < \frac{1}{2}\] ?

Q 17 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{2^{x + 1}}{1 - 4^x} \right), - \infty < x < 0\] ?

Q 18 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{2 a^x}{1 - a^{2x}} \right), a > 1, - \infty < x < 0\] ?

Q 19 | Page 63

Differentiate \[\sin^{- 1} \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} \right\}, 0 < x < 1\] ?

Q 20 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?

Q 21 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi\] ?

Q 22 | Page 63

Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?

Q 23 | Page 63

Differentiate \[\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty\] ?

Q 24 | Page 63

Differentiate \[\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R\] ?

Q 25 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?

Q 26 | Page 63

Differentiate  \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?

Q 27 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?

Q 28 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?

Q 29 | Page 63

 Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?

Q 30 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?

Q 31 | Page 64

Differentiate \[\tan^{- 1} \left( \frac{5 x}{1 - 6 x^2} \right), - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}\] ?

Q 32 | Page 63

Differentiate 

\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?

Q 33 | Page 64

Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?

Q 34 | Page 64

Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?

Q 35 | Page 64

If  \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), 0 < x < 1,\] prove that  \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?

 

Q 36 | Page 64

If \[y = \sin^{- 1} \left( \frac{x}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right), 0 < x < \infty\] prove that  \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?

 

Q 37 | Page 64

Differentiate the following with respect to x

\[\left( i \right) \cos^{- 1} \left( \sin x \right)\]

(ii)  \[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]

Q 38 | Page 64

If  \[y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}\],  show that \[\frac{dy}{dx}\] is independent of x. ? 

 

Q 39 | Page 64

If \[y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0\] ,prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ? 

Q 40 | Page 64

If  \[y = se c^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right), x > 0 . \text{ Find} \frac{dy}{dx}\] ?

 

Q 41 | Page 64

If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?

Q 42 | Page 64

If  \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, 0 < x < \frac{1}{2}, \text{ find } \frac{dy}{dx} .\] ?

Q 43 | Page 64

If the derivative of tan−1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b ?

Q 44 | Page 64

If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, - \frac{1}{2} < x < 0, \text{ find } \frac{dy}{dx} \] ?

Q 45 | Page 64

If \[y = \tan^{- 1} \left( \frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}} \right), \text{find } \frac{dy}{dx}\] ?

Q 46 | Page 64

If \[y = \cos^{- 1} \left\{ \frac{2x - 3 \sqrt{1 - x^2}}{\sqrt{13}} \right\}, \text{ find } \frac{dy}{dx}\] ?

Q 47 | Page 64

Differentiate \[\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left( 36 \right)^x} \right\}\]  with respect to x ?

Q 48 | Page 64

If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?

Pages 74 - 75

Q 1 | Page 74

Find \[\frac{dy}{dx}\] in each of the following cases \[xy = c^2\]  ?

Q 2 | Page 74

Find  \[\frac{dy}{dx}\] in each of the following cases \[y^3 - 3x y^2 = x^3 + 3 x^2 y\] ?

 

Q 3 | Page 74

Find   \[\frac{dy}{dx}\] in each of the following cases  \[x^{2/3} + y^{2/3} = a^{2/3}\] ?

 

Q 4 | Page 74

Find  \[\frac{dy}{dx}\] in each of the following cases  \[4x + 3y = \log \left( 4x - 3y \right)\] ?

 

Q 5 | Page 74

Find  \[\frac{dy}{dx}\] in each of the following cases  \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?

 

Q 6 | Page 74

Find  \[\frac{dy}{dx}\] in each of the following cases \[x^5 + y^5 = 5 xy\] ?

 

Q 7 | Page 74

Find  \[\frac{dy}{dx}\] in each of the following cases \[\left( x + y \right)^2 = 2axy\] ?

 

Q 8 | Page 74

Find  \[\frac{dy}{dx}\] in each of the following cases \[\left( x^2 + y^2 \right)^2 = xy\] ?

 

Q 9 | Page 74

Find  \[\frac{dy}{dx}\] in each of the following cases \[\tan^{- 1} \left( x^2 + y^2 \right) = a\] ?

 

Q 10 | Page 74

Find  \[\frac{dy}{dx}\] in each of the following cases \[e^{x - y} = \log \left( \frac{x}{y} \right)\] ?

 

Q 11 | Page 74

Find  \[\frac{dy}{dx}\] in each of the following cases \[\sin xy + \cos \left( x + y \right) = 1\] ?

 

Q 12 | Page 74

If \[\sqrt{1 - x^2} + \sqrt{1 - y^2} = a \left( x - y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{1 - x^2}\] ?

Q 13 | Page 75

If \[y \sqrt{1 - x^2} + x \sqrt{1 - y^2} = 1\] ,prove that \[\frac{dy}{dx} = - \sqrt{\frac{1 - y^2}{1 - x^2}}\] ?

Q 14 | Page 75

If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?

Q 15 | Page 75

If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?

Q 16 | Page 75

If \[x \sqrt{1 + y} + y \sqrt{1 + x} = 0\] , prove that \[\left( 1 + x \right)^2 \frac{dy}{dx} + 1 = 0\]  ?

Q 17 | Page 75

If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?

Q 18 | Page 75

If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that  \[\frac{dy}{dx} = \frac{y}{x}\] ?

Q 19 | Page 75

If \[\tan^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = a\] Prove that  \[\frac{dy}{dx} = \frac{x}{y}\frac{\left( 1 - \tan a \right)}{\left( 1 + \tan a \right)}\] ?

Q 20 | Page 75

If \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?

Q 21 | Page 75

If \[y = x \sin \left( a + y \right)\] ,Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?

Q 22 | Page 75

If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?

Q 23 | Page 75

If \[y = x \sin y\] , Prove that \[\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}\] ?

Q 24 | Page 75

If \[y \sqrt{x^2 + 1} = \log \left( \sqrt{x^2 + 1} - x \right)\] ,Show that \[\left( x^2 + 1 \right) \frac{dy}{dx} + xy + 1 = 0\] ?

Q 25 | Page 75

If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find}  \frac{dy}{dx}\] ?

Q 26 | Page 75

If  \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find}  \frac{dy}{dx}\] ?

Q 27 | Page 75

\[If e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?

Q 28 | Page 75

If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?

Q 29 | Page 75

If \[\sin^2 y + \cos xy = k,\] find  \[\frac{dy}{dx}\] at \[x =\] \[y = \frac{\pi}{4} .\] ?

Q 30 | Page 75

If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?

Q 31 | Page 75

If \[\sqrt{y + x} + \sqrt{y - x} = c, \text {show that } \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}\] ?

Pages 88 - 90

Q 1 | Page 88

Differentiate \[x^{1/x}\] ?

Q 2 | Page 88

Differentiate \[x^{\sin x}\]  ?

Q 3 | Page 88

Differentiate \[\left( 1 + \cos x \right)^x\] ?

Q 4 | Page 88

Differentiate \[x^{\cos^{- 1} x}\] ?

Q 5 | Page 88

Differentiate \[\left( \log x \right)^x\] ?

Q 6 | Page 88

Differentiate \[\left( \log x \right)^{\cos x}\] ?

Q 7 | Page 88

Differentiate \[\left( \sin x \right)^{\cos x}\] ?

Q 8 | Page 88

Differentiate \[e^{x \log x}\] ?

Q 9 | Page 88

Differentiate  \[\left( \sin x \right)^{\log x}\] ?

Q 10 | Page 88

Differentiate \[{10}^{ \log \sin x }\] ?

Q 11 | Page 88

Differentiate \[\left( \log x \right)^{ \log x }\] ?

Q 12 | Page 88

Differentiate \[{10}^\left( {10}^x \right)\] ?

Q 13 | Page 88

Differentiate  \[\sin \left( x^x \right)\] ?

Q 14 | Page 88

Differentiate \[\left( \sin^{- 1} x \right)^x\] ?

Q 15 | Page 88

Differentiate \[x^{\sin^{- 1} x}\]  ?

Q 16 | Page 88

Differentiate \[\left( \tan x \right)^{1/x}\] ?

Q 17 | Page 88

Differentiate \[x^{\tan^{- 1} x }\]  ?

Q 18.1 | Page 88

Differentiate  \[\left( x^x \right) \sqrt{x}\] ?

Q 18.2 | Page 88

Differentiate \[x^\left( \sin x - \cos x \right) + \frac{x^2 - 1}{x^2 + 1}\] ?

Q 18.3 | Page 88

Differentiate  \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\]  ?

Q 18.4 | Page 88

Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?

Q 18.5 | Page 88

Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?

Q 18.6 | Page 88

Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?

Q 18.7 | Page 88

Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?

Q 18.8 | Page 88

Differentiate  \[x^{x^2 - 3} + \left( x - 3 \right)^{x^2}\] ?

Q 19 | Page 89

Find  \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?

 

Q 20 | Page 89
Find \[\frac{dy}{dx}\]  \[y = x^n + n^x + x^x + n^n\] ?
Q 21 | Page 89

find  \[\frac{dy}{dx}\]  \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?

 

Q 22 | Page 89

Find  \[\frac{dy}{dx}\]  \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?

 

Q 23 | Page 89

Find  \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?

 

Q 24 | Page 89

Find  \[\frac{dy}{dx}\] \[y = \sin x \sin 2x \sin 3x \sin 4x\] ?

 

Q 25 | Page 89

Find  \[\frac{dy}{dx}\] \[y = x^{\sin x} + \left( \sin x \right)^x\] ?

Q 26 | Page 89

Find  \[\frac{dy}{dx}\]  \[y = \left( \sin x \right)^{\cos x} + \left( \cos x \right)^{\sin x}\] ?

 

Q 27 | Page 89

Find \[\frac{dy}{dx}\] \[y =  \left( \tan  x \right)^{\cot }  x  +  \left( \cot  x \right)^{\tan  x}\] ?

Q 28 | Page 89

Fine \[\frac{dy}{dx}\] \[y = \left( \sin x \right)^x + \sin^{- 1} \sqrt{x}\] ?

Q 29.1 | Page 89

Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?

Q 29.2 | Page 89

Find \[\frac{dy}{dx}\]  \[y = x^x + \left( \sin x \right)^x\] ?

Q 30 | Page 89

Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?

Q 31 | Page 89

Find \[\frac{dy}{dx}\] \[y = x^x + x^{1/x}\] ?

Q 32 | Page 89

Find \[\frac{dy}{dx}\] \[y = x^{\log x }+ \left( \log x \right)^x\] ?

Q 33 | Page 89

If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?

Q 34 | Page 89

If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?

Q 35 | Page 89

If \[y = \sin \left( x^x \right)\] prove that  \[\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)\] ?

Q 36 | Page 89

If \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?

Q 37 | Page 89

If \[x^y \cdot y^x = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}\] ?

Q 38 | Page 89

If \[x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}\] ?

Q 39 | Page 89

If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?

Q 40 | Page 89

If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?

Q 41 | Page 89

If \[\left( \sin x \right)^y = \left( \cos y \right)^x ,\], prove that \[\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}\] ?

Q 42 | Page 89

If \[\left( \cos x \right)^y = \left( \tan y \right)^x\] , prove that \[\frac{dy}{dx} = \frac{\log \tan y + y \tan x}{ \log \cos x - x \sec y \ cosec\ y }\] ?

Q 43 | Page 89

If \[e^x + e^y = e^{x + y}\] , prove that

\[\frac{dy}{dx} + e^{y - x} = 0\] ?

Q 44 | Page 90

If \[e^y = y^x ,\] prove that\[\frac{dy}{dx} = \frac{\left( \log y \right)^2}{\log y - 1}\] ?

Q 45 | Page 90

If \[e^{x + y} - x = 0\] ,prove that \[\frac{dy}{dx} = \frac{1 - x}{x}\] ?

Q 46 | Page 90

If \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?

 

Q 47 | Page 90

If  \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?

 

Q 48 | Page 90

If  \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?

 

Q 49 | Page 90

If \[xy \log \left( x + y \right) = 1\] , prove that  \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?

Q 50 | Page 90

If \[y = x \sin y\] , prove that  \[\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}\] ?

 

Q 51 | Page 90

Find the derivative of the function f (x) given by  \[f\left( x \right) = \left( 1 + x \right) \left( 1 + x^2 \right) \left( 1 + x^4 \right) \left( 1 + x^8 \right)\] and hence find f' (1) ?

 

Q 52 | Page 90

If \[y = \log\frac{x^2 + x + 1}{x^2 - x + 1} + \frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{\sqrt{3} x}{1 - x^2} \right), \text{ find } \frac{dy}{dx} .\] ?

Q 53 | Page 90

If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?

Q 54 | Page 90

If  \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?

 

Q 55 | Page 90

If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?

Q 56 | Page 90

If  \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?

 

Q 57 | Page 90
\[\text{ If }\cos y = x\cos\left( a + y \right),\text{  where } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
Q 58 | Page 90
\[\text{ If } \left( x - y \right) e^\frac{x}{x - y} = a,\text{  prove that y }\frac{dy}{dx} + x = 2y\] ?
Q 59 | Page 90
\[\text{ If } x = e^{x/y} , \text{ prove that } \frac{dy}{dx} = \frac{x - y}{x\log x}\] ?
Q 60 | Page 90
\[\text{ If }y = x^{\tan x} + \sqrt{\frac{x^2 + 1}{2}}, \text{ find} \frac{dy}{dx}\] ?

 

Q 61 | Page 90
\[\text{ If y } = 1 + \frac{\alpha}{\left( \frac{1}{x} - \alpha \right)} + \frac{{\beta}/{x}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)} + \frac{{\gamma}/{x^2}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)\left( \frac{1}{x} - \gamma \right)}, \text{ find } \frac{dy}{dx} \] ?

Pages 98 - 99

Q 1 | Page 98

If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?

Q 2 | Page 98

If \[y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + . . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sin x}{1 - 2 y}\] ?

Q 3 | Page 98

If  \[y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + . . to \infty}}}\]  \[\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}\] ?

 

Q 4 | Page 98

If  \[y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sec^2 x}{2 y - 1}\] ?

 

Q 5 | Page 98

\[y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}} \],prove that \[\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}\] ?

Q 6 | Page 98

If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?

 

Q 7 | Page 99

If \[y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}\], prove that  \[\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}\]\[+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}\] ?

 

Q 8 | Page 99

If \[y = \left( \cos x \right)^{\left( \cos x \right)^{\left( \cos x \right) . . . \infty}}\],prove that \[\frac{dy}{dx} = - \frac{y^2 \tan x}{\left( 1 - y \log \cos x \right)}\]?

 

Pages 103 - 104

Q 1 | Page 103

Find \[\frac{dy}{dx}\] \[x = a t^2 \text{ and } y = 2\ at \] ?

Q 2 | Page 103

Find \[\frac{dy}{dx}\], When \[x = a \left( \theta + \sin \theta \right) \text{ and } y = a \left( 1 - \cos \theta \right)\] ?

Q 3 | Page 103

If \[\frac{dy}{dx}\] when \[x = a \cos \theta \text{ and } y = b \sin \theta\] ?

Q 4 | Page 103

Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?

Q 5 | Page 103

Find \[\frac{dy}{dx}\] , when \[x = b   \sin^2   \theta  \text{ and }  y = a   \cos^2   \theta\] ?

Q 6 | Page 103

Find \[\frac{dy}{dx}\] ,When \[x = a \left( 1 - \cos \theta \right) \text{ and } y = a \left( \theta + \sin \theta \right) \text{ at } \theta  = \frac{\pi}{2}\] ?

Q 7 | Page 103

Find \[\frac{dy}{dx}\] ,when \[x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\] ?

Q 8 | Page 103

Find \[\frac{dy}{dx}\] , when \[x = \frac{3 at}{1 + t^2}, \text{ and } y = \frac{3 a t^2}{1 + t^2}\] ?

Q 9 | Page 103

Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?

Q 10 | Page 103

Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?

Q 11 | Page 103

Find \[\frac{dy}{dx}\] when \[x = \frac{2 t}{1 + t^2} \text{ and } y = \frac{1 - t^2}{1 + t^2}\] ?

Q 12 | Page 103

Find \[\frac{dy}{dx}\] , when  \[x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R\] ?

Q 13 | Page 103

Find  \[\frac{dy}{dx}\] , when  \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?

 

Q 14 | Page 103

If  \[x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta\]  \[\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)\] ?

Q 15 | Page 103

If \[x = e^{\cos 2 t} \text{ and y }= e^{\sin 2 t} ,\] prove that \[\frac{dy}{dx} = - \frac{y \log x}{x \log y}\] ?

Q 16 | Page 103

If \[x = \cos t \text{ and y }  = \sin t,\] prove that  \[\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}\] ?

 

Q 17 | Page 103

If  \[x = a\left( t + \frac{1}{t} \right) \text{ and y } = a\left( t - \frac{1}{t} \right)\] ,prove that  \[\frac{dy}{dx} = \frac{x}{y}\]?

 

Q 18 | Page 103
If \[x = \sin^{- 1} \left( \frac{2 t}{1 + t^2} \right) \text{ and y } = \tan^{- 1} \left( \frac{2 t}{1 - t^2} \right), - 1 < t < 1\] porve that \[\frac{dy}{dx} = 1\] ?

 

Q 19 | Page 103

If  \[x = \frac{\sin^3 t}{\sqrt{\cos 2 t}}, y = \frac{\cos^3 t}{\sqrt{\cos t 2 t}}\] , find\[\frac{dy}{dx}\] ?

 

Q 20 | Page 103

If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?

Q 21 | Page 103

If \[x = a \left( \frac{1 + t^2}{1 - t^2} \right) \text { and y } = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}\] ?

Q 22 | Page 104

If \[x = 10 \left( t - \sin t \right), y = 12 \left( 1 - \cos t \right), \text { find } \frac{dy}{dx} .\] ?

 

Q 23 | Page 104

If \[x = a \left( \theta - \sin \theta \right) and, y = a \left( 1 + \cos \theta \right), \text { find } \frac{dy}{dx} at \theta = \frac{\pi}{3} \] ?

 

Q 24 | Page 104

If  \[x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)\] , show that at  \[t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}\] ?

Q 25 | Page 104

\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx} at t = \frac{\pi}{4}\] ?

Q 26 | Page 104

If  \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?

Q 27 | Page 104
\[\sin x = \frac{2t}{1 + t^2}, \tan y = \frac{2t}{1 - t^2}, \text { find }  \frac{dy}{dx}\] ?
Q 28 | Page 104

Write the derivative of sinx with respect to cosx ?

Pages 112 - 113

Q 1 | Page 112

Differentiate x2 with respect to x3

Q 2 | Page 112

Differentiate log (1 + x2) with respect to tan−1 x ?

Q 3 | Page 112

Differentiate (log x)x with respect to log x ?

Q 4.1 | Page 112

Differentiate  \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\]\[x \in \left( 0, 1 \right)\]  ?

 

Q 4.2 | Page 112

Differentiate  \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\] \[x \in \left( - 1, 0 \right)\] ?

Q 5.1 | Page 112
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2 \sqrt{2}}, \frac{1}{\sqrt{2 \sqrt{2}}} \right)\] ?
Q 5.2 | Page 112

Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)\] ?

Q 5.3 | Page 112

Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2}, - \frac{1}{2 \sqrt{2}} \right)\] ?

Q 6 | Page 112

Differentiate\[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)\] with respect to \[\sin^{-1} \left( \frac{2x}{1 + x^2} \right)\], If \[- 1 < x < 1, x \neq 0 .\] ?

Q 7.1 | Page 112

Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to  \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\] \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?

Q 7.2 | Page 112

Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to  \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\] \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?

Q 8 | Page 112

Differentiate \[\left( \cos x \right)^{\sin x }\] with respect to \[\left( \sin x \right)^{\cos x }\]?

Q 9 | Page 112

Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { if } 0 < x < 1\] ?

Q 10 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{1 + ax}{1 - ax} \right)\] with respect to \[\sqrt{1 + a^2 x^2}\] ?

Q 11 | Page 113

Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?

Q 12 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text {  if }0 < x < 1\] ?

Q 13 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{x - 1}{x + 1} \right)\] with respect to \[\sin^{- 1} \left( 3x - 4 x^3 \right), \text { if }- \frac{1}{2} < x < \frac{1}{2}\] ?

Q 14 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with  respect to \[\sec^{- 1} x\] ?

Q 15 | Page 113

Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text{ if } - 1 < x < 1\] ?

Q 16 | Page 113

Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), if \frac{1}{2} < x < 1\] ? 

Q 17 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)\] with respect to \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right), \text { if } - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?

Q 18 | Page 113

\[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cot^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right),\text { if }0 < x < 1\] ? 

Q 19 | Page 113

Differentiate \[\sin^{- 1} \left( 2 ax \sqrt{1 - a^2 x^2} \right)\] with respect to \[\sqrt{1 a^2 x^2}, if - \frac{1}{\sqrt{2}} < ax < \frac{1}{\sqrt{2}}\] ?

Q 20 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)\] with respect to \[\sqrt{1 - x^2},\text {if} - 1 < x < 1\] ?

Pages 116 - 118

Q 1 | Page 117

If f (x) = loge (loge x), then write the value of f' (e) ?

Q 2 | Page 117

If \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?

Q 3 | Page 117

If \[f'\left( 1 \right) = 2 \text { and y } = f \left( \log_e x \right), \text { find} \frac{dy}{dx} \text { at }x = e\] ?

Q 4 | Page 117

If \[f\left( 1 \right) = 4, f'\left( 1 \right) = 2\] find the value of the derivative of  \[\log \left( f\left( e^x \right) \right)\] w.r. to x at the point x = 0 ?

 

Q 5 | Page 117

If \[f'\left( x \right) = \sqrt{2 x^2 - 1} \text { and y } = f \left( x^2 \right)\] then find \[\frac{dy}{dx} \text { at } x = 1\] ?

Q 6 | Page 117

Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and f' (3) = 9, write the value of g' (9).

Q 7 | Page 117

If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?

Q 8 | Page 117

If \[\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx} \] ?

Q 9 | Page 117

If \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?

Q 10 | Page 118

If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?

Q 11 | Page 118

If \[f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}\] write the value of \[\frac{dy}{dx} \text{ at x } = 0\] ?

Q 12 | Page 118

If \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?

Q 13 | Page 118

If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?

Q 14 | Page 118

If \[x = a \left( \theta + \sin \theta \right), y = a \left( 1 + \cos \theta \right), \text{ find} \frac{dy}{dx}\] ?

Q 15 | Page 118

If \[- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}\] ?

Q 16 | Page 116

If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?

Q 17 | Page 118

If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\]  ?

Q 18 | Page 118

If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ? 

Q 19 | Page 118

If \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?

Q 20 | Page 118

If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text{ find } \frac{dy}{dx}\] ?

Q 21 | Page 118

If \[y = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right)\] then write the value of \[\frac{dy}{dx} \] ?

Q 22 | Page 118

If \[\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . . \]  to ∞, then find the value of  \[\frac{dy}{dx}\] ?

Q 23 | Page 118

If \[u = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ and v} = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] where \[- 1 < x < 1\], then write the value of \[\frac{du}{dv}\] ?

Q 24 | Page 118

If \[f\left( x \right) = \log \left\{ \frac{u \left( x \right)}{v \left( x \right)} \right\}, u \left( 1 \right) = v \left( 1 \right) \text{ and }u' \left( 1 \right) = v' \left( 1 \right) = 2\] , then find the value of f' (1) ?

Q 25 | Page 118

If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ? 

Q 26 | Page 118

If f (x) is an even function, then write whether f' (x) is even or odd ?

Q 27 | Page 118

If f (x) is an odd function, then write whether f' (x) is even or odd ?

Q 28 | Page 118

If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?

Pages 119 - 122

Q 1 | Page 119

If f (x) = logx2 (log x), the f' (x) at x = e is
(a) 0
(b) 1
(c) 1/e
(d) 1/2 e

Q 2 | Page 119

The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is 

(a) \[\frac{x}{\log x}\]

(b)  \[\frac{\log x}{x}\]

(c) \[\left( x \log x \right)^{- 1}\]

(d) none of these

 

Q 3 | Page 119

The derivative of the function

\[\cot^{- 1} \left| \left( \cos 2 x \right)^{1/2} \right| \text{ at } x = \pi/6 \text{ is }\]
(a) (2/3)1/2
(b) (1/3)1/2
(c) 31/2
(d) 61/2
Q 4 | Page 119

Differential coefficient of sec

\[\sec \left( \tan^{- 1} x \right)\] is
(a)  \[\frac{x}{1 + x^2}\]
(b)  \[x \sqrt{1 + x^2}\]
(c) \[\frac{1}{\sqrt{1 + x^2}}\]
(d) \[\frac{x}{\sqrt{1 + x^2}}\]

 

Q 5 | Page 119

If \[f\left( x \right) = \tan^{- 1} \sqrt{\frac{1 + \sin x}{1 - \sin x}}, 0 \leq x \leq \pi/2, \text{ then } f' \left( \pi/6 \right) \text{ is }\]

(a) − 1/4
(b) − 1/2
(c) 1/4
(d) 1/2

Q 6 | Page 119

If \[y = \left( 1 + \frac{1}{x} \right)^x , \text{ then} \frac{dy}{dx} =\]

(a) \[\left( 1 + \frac{1}{x} \right)^x \left( 1 + \frac{1}{x} \right) - \frac{1}{x + 1}\]

(b) \[\left( 1 + \frac{1}{x} \right)^x \log \left( 1 + \frac{1}{x} \right)\]

(c) \[\left( x + \frac{1}{x} \right)^x \left\{ \log \left( x + 1 \right) - \frac{x}{x + 1} \right\}\]

(d) \[\left( x + \frac{1}{x} \right)^x \left\{ \log \left( 1 + \frac{1}{x} \right) + \frac{1}{x + 1} \right\}\]

Q 7 | Page 119

If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is 

(a) \[\frac{1 + x}{1 + \log x}\]

(b) \[\frac{1 - \log x}{1 + \log x}\]

(c) not defined

(d) \[\frac{\log x}{\left( 1 + \log x \right)^2}\]

Q 8 | Page 119

Given  \[f\left( x \right) = 4 x^8 , \text { then }\]

(a) \[f'\left( \frac{1}{2} \right) = f'\left( - \frac{1}{2} \right)\]

(b) \[f\left( \frac{1}{2} \right) = - f'\left( - \frac{1}{2} \right)\]

(c) \[f\left( - \frac{1}{2} \right) = f\left( - \frac{1}{2} \right)\]

(d) \[f\left( \frac{1}{2} \right) = f'\left( - \frac{1}{2} \right)\]

Q 9 | Page 119

If \[x = a \cos^3 \theta, y = a \sin^3 \theta, \text { then } \sqrt{1 + \left( \frac{dy}{dx} \right)^2} =\]

(a) \[\tan^2 \theta\]

(b) \[\sec^2 \theta\]

(c) \[\sec \theta\]

(d) \[\left| \sec \theta \right|\]

Q 10 | Page 120

If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\]

(a) \[- \frac{2}{1 + x^2}\]

(b) \[\frac{2}{1 + x^2}\]

(c) \[\frac{1}{2 - x^2}\]

(d) \[\frac{2}{2 - x^2}\]

Q 11 | Page 120

The derivative of \[\sec^{- 1} \left( \frac{1}{2 x^2 + 1} \right) \text { w . r . t }. \sqrt{1 + 3 x} \text { at } x = - 1/3\]

(a) does not exist
(b) 0
(c) 1/2
(d) 1/3

Q 12 | Page 120

For the curve \[\sqrt{x} + \sqrt{y} = 1, \frac{dy}{dx}\text {  at } \left( 1/4, 1/4 \right)\text {  is }\]

(a) 1/2
(b) 1
(c) −1
(d) 2
Q 13 | Page 120

If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\]

(a) 2
(b) − 2
(c) 1
(d) − 1]

 

Q 14 | Page 120

Let  \[\cup = \sin^{- 1} \left( \frac{2 x}{1 + x^2} \right) \text { and }V = \tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text { then } \frac{d \cup}{dV} =\]

(a) 1/2
(b) x
(c) \[\frac{1 - x^2}{x^2 - 4}\]

(d) 1

Q 15 | Page 120

\[\frac{d}{dx} \left\{ \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right) \right\} \text { equals }\]

(a) 1/2
(b) − 1/2
(c) 1
(d) − 1

 

Q 16 | Page 120
\[\frac{d}{dx} \left[ \log \left\{ e^x \left( \frac{x - 2}{x + 2} \right)^{3/4} \right\} \right]\] equals 
(a) \[\frac{x^2 - 1}{x^2 - 4}\]
(b) 1
(c)\[\frac{x^2 + 1}{x^2 - 4}\]
(d)  \[e^x \frac{x^2 - 1}{x^2 - 4}\]
Q 17 | Page 120

If \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\]

(a) \[\frac{\sin x}{2 y - 1}\]

(b) \[\frac{\sin x}{1 - 2 y}\]

(c) \[\frac{\cos x}{1 - 2 y}\]

(d) \[\frac{\cos x}{2 y - 1}\]

Q 18 | Page 120

If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\]

(a) \[- \frac{y}{x}\]

(b) \[\frac{3 \sin \left( xy \right) + 4 \cos \left( xy \right)}{3 \cos \left( xy \right) - 4 \sin \left( xy \right)}\]

(c) \[\frac{3 \cos \left( xy \right) + 4 \sin \left( xy \right)}{4 \cos \left( xy \right) - 3 \sin \left( xy \right)}\]

(d) none of these

 

Q 19 | Page 120

If \[\sin y = x \sin \left( a + y \right), \text { then }\frac{dy}{dx} \text { is}\]

(a) \[\frac{\sin a}{\sin a \sin^2 \left( a + y \right)}\]

(b) \[\frac{\sin^2 \left( a + y \right)}{\sin a}\]

(c) \[\sin a \sin^2 \left( a + y \right)\]

(d) \[\frac{\sin^2 \left( a - y \right)}{\sin a}\]

Q 20 | Page 121

The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to  \[\cos^{- 1} x\]  is 

(a) 2

(b) \[\frac{1}{2 \sqrt{1 - x^2}}\]

(c) \[2/x\]

(d) \[1 - x^2\]

Q 21 | Page 121

If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to

(a) \[1 \text { for x } < - 3\]

(b) \[- 1\text {  for x} < - 3\]

(c) \[1\text {  for all } x \in R\]

(d) none of these

Q 22 | Page 121

If \[f\left( x \right) = \left| x^2 - 9x + 20 \right|\]  then f' (x) is equal to 

(a) \[- 2x + 9\text {  for all } x \in R\]

(b) \[2x - 9 \text { if }4 < x < 5\]

(c) \[- 2x + 9, \text { if }4 < x < 5\]

(d) none of these

Q 23 | Page 121

If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\]  then the derivative of f (x) in the interval [0, 7] is

(a) 1
(b) −1
(c) 0
(d) none of these

Q 24 | Page 121

If \[f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)\]  is equal to 

(a) 1
(b) −1
(c) 0
(d) none of these

Q 25 | Page 121

If \[f\left( x \right) = \left( \frac{x^l}{x^m} \right)^{l + m} \left( \frac{x^m}{x^n} \right)^{m + n} \left( \frac{x^n}{x^l} \right)^{n + 1}\] the f' (x) is equal to

(a) 1
(b) 0
(c) \[x^{l + m + n}\]

(d) none of these

Q 26 | Page 121

If \[y = \frac{1}{1 + x^{a - b} +^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} + \frac{1}{1 + x^{b - a} + x^{c - a}}\] then \[\frac{dy}{dx}\]  is equal to

(a) 1
(b) \[\left( a + b + c \right)^{x^{a + b + c - 1}}\]
(c) 0
(d) none of these
Q 27 | Page 121

If  \[\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)\] then \[\frac{dy}{dx}\] is equal to 

(a) \[\frac{x^2}{y^2} \sqrt{\frac{1 - y^6}{1 - x^6}}\]

(b) \[\frac{y^2}{x^2}\sqrt{\frac{1 - y^6}{1 + x^6}}\]

(c) \[\frac{x^2}{y^2}\sqrt{\frac{1 - x^6}{1 - y^6}}\]

(d) none of these

Q 28 | Page 121

If \[y = \log \sqrt{\tan x}\] then the value of

\[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by

(a) ∞
(b) 1
(c) 0
(d) \[\frac{1}{2}\]

Q 29 | Page 121

If \[\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}\] is equal to

(a) \[\frac{x^2 - y^2}{x^2 + y^2}\]

(b)  \[\frac{y}{x}\]

(c) \[\frac{x}{y}\]

(d) none of these

Q 30 | Page 121

If \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to

(a) \[\frac{\cos^2 \left( a + y \right)}{\cos a}\]

(b)\[\frac{\cos a}{\cos^2 \left( a + y \right)}\]

(c) \[\frac{\sin^2 y}{\cos a}\]

(d) none of these

Q 31 | Page 122

If \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\]

(a) \[\frac{4 x^3}{1 - x^4}\]

(b) \[- \frac{4x}{1 - x^4}\]

(c) \[\frac{1}{4 - x^4}\]

(d) \[- \frac{4 x^3}{1 - x^4}\]

Q 32 | Page 122

If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\]

(a) \[\frac{\cos x}{2y - 1}\]

(b) \[\frac{\cos x}{1 - 2y}\]

(c)  \[\frac{\sin x}{1 - 2y}\]

(d) \[\frac{\sin x}{2y - 1}\]

Q 33 | Page 122

If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then  } \frac{dy}{dx}\] is equal to

(a) \[\frac{1}{2}\]

(b) 0
(c) 1
(d) none of these

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

RD Sharma solutions for Class 12 Mathematics chapter 11 - Differentiation

RD Sharma solutions for Class 12 Maths chapter 11 (Differentiation) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 11 Differentiation are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

Using RD Sharma Class 12 solutions Differentiation exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

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